Saturday, February 6, 2016

Lessons from "Lessons from Bowen and Darryl"

I had the beginnings of a blinding insight this week that I wanted to write down and think about. It all started with Ben Blum-Smith’s blog post about what he learned from Bowen and Darryl’s master class at JMM earlier this year. He wrote up his takeaways and it sparked many great Twitter conversations.

I wanted to write about what *I* took away from his post and what I have learned from subsequent Twitter conversations with Bowen and Darryl.

My burning question this week that I posed for them after reading Ben’s post was, how do they manage the mixture of “speed demons” and “katamari” in their class work. This distinction between speed demons and katamari was really obvious once I read about it; yet, it remains the dirty little secret of math pedagogy — and the unspoken, problematic truth of group work strategies.

Both of these approaches—the “speed demon” approach and the “katamari” approach —  are highly personal, highly developed inner world views within mindsets. They are not labels or designations applied from without. They are basic “come-from” attitudes that arise from within.

I have my own truth of this distinction from my own experiences, and I believe that we all approach it with our own biases and theoretical frameworks. I freely admit that I speak as a katamari with a lifetime of bad experiences in mathematical group work, both with fellow students and with professional colleagues. In mathematics and in group work, I find that I still experience these hidden assumptions over and over again: what I perceive as the tyranny of the speed demons and my own resigned sense of hopelessness that my tortoise-like katamari learning style — including my tortoise-shell defense mechanisms against the feelings of rage and powerlessness and inner worthlessness as a math learner I experience whenever I am asked to do mathematical learning in a group with others.
[EDITORIAL NOTE: Please don’t worry that you need to rescue me. Or that I need you to rescue me. I don’t and you don’t. These are only thoughts flowing down the river of mind, and I have learned how to notice them and work with them through many years of self-noticing, meditation, inner development work, and therapy. I’m not held back by them, and I don’t need to be reassured about them. But the reality of inner development is that those deep-rooted holes and feelings don’t go away. We just learn how to notice them and work with them more skillfully and artfully so we can continue under all circumstances. Over time, they lose their power. They just become the voice of monkey mind yakking away in the background. And I have learned how to work with that noise in the background and to tune it out].
For math learners with a high degree of natural aptitude and curiosity, the speed demon world view is a very natural attitude to develop. You love math, you want more of it, and you are totally and completely voracious. I see this in many of my students. They want to devour math. They’re not just hungry, they are  completely driven by their appetite for math. More more more. Faster faster faster. Nom nom nom. They find math delicious, and they want to eat all they can as fast as they can. They want to climb every mountain they encounter. The higher, the better. More mountains, please. More math.

They’re joyful — they’re not intentionally being aggressive in the classroom. But they are children. They don’t have a lot of self-control or self-regulation skills, and they’re adolescents, so they don’t have a lot of awareness of how others are feeling in the moment. That is why they have to be managed and fed in the classroom learning process.

Meanwhile, the katamari in my classes are experiencing things quite differently. But first, we should mention that there is a HUGE range of ability, interest, and aptitude among the katamari. In fact, the population of katamari is where the greatest range of learners exists. But they are distinguished by their katamari worldview, which arises in binary opposition to the speed demon worldview. They work through things at their own pace and they alternate between individual think time moments and collaboration. And they build as patiently as they can on what they figure out.

Bowen and Darryl propose a revolutionary approach to managing the mixture of speed demons and katamari within the problem-based, group-work-centric math class. They divide classroom time into “doing” segments and whole-class “discussing” segments.

Here is my summary understanding of the three key parts of their deployment strategy for the problem-based learning experience.
1. Problem sets are designed to be a treasure map — but the essential treasure in any day’s work is located within the “Important Stuff” initial section
2. Groupings must facilitate individual discovery together — and they must eliminate all social and emotional obstacles to including *everybody* in the process of discovery See #1.
3. Whole-class discussion segments exist only to feature the discoveries that katamari have made by slowing down and really noticing the often subtle and deep mathematics that can be noticed through careful work
Some elaboration on all of this:


The daily problem set is designed as a treasure map, but the secret is that all essential mathematics in any day’s work is located in the first section, not at the end of the problem set.

Interesting Stuff and Tough Stuff exist to provide nourishment for anyone who is ready to explore it. But it does NOT contain the keys to the kingdom. If speed demons wish to zoom ahead and tackle the “tough stuff” that is there to satisfy their appetite for zooming into zoomy, zoomy heights, then they are welcome to do so. Meanwhile, the katamari can find reassurance in the fact that their worldview and their approach is explicitly being valued.


This point is key. Our students are still adolescents, and they are starting from a place of little impulse control when it comes to their own self-interest. Since the only way to develop intrinsic motivation is through autonomy, mastery, and purpose, we need to tap into that rather than try to manipulate everybody into clamping down on their natural orientations.

Since what motivates people is a combination of autonomy, mastery, and purpose (see Dan Pink, Drive), our grouping strategies in the classroom HAVE TO support student autonomy. In other words, if speed demons believe they’ve gotta speed, then our groupings need to support their desire/need for speed.

It’s important to separate the speed demons from the katamari, but there is a crucial misunderstanding as to why. Many people believe that this places an undue burden on the speed demons, but that’s actually backwards from the truth. The reality is, mixing the groups during the “doing” segments actually places an unfair burden on the katamari. It requires them to allow the speed demons to dominate the learning process, to be the center of attention at all times, and to cheat them out of their understanding.

So for this reason, it’s important to let the speed demons go off in zoomy groups and zoom away. This is not the place where they are going to learn the social and emotional interpersonal skills they need because it’s NOT the place where they are receptive to these lessons. Instead, this is the place where we, the adults, need to create safe space for katamari to work at their own pace and to develop the learnings they need in order to move forward.

*This* is meaningful differentiation.

So during the doing segments, I am now going to let the speed demons zoom. In fact, I am going to set up my speed demons so they can go off and do their zoomy zoomy zoom investigations with the “Interesting Stuff” and the “Tough Stuff” in the daily problem sets.


This strategy allows for a wonderfully cross-pollinating atmosphere to arise in whole-class discussion segments. Since everyone has received what they individually needed during the “doing” segments, they are now free to be more open and receptive to what others experienced and discovered while they were lost in their own autonomous worldview.

They can also pay attention to what the instructors really want everybody in the room to experience.

What I value about this strategy is how it turns whole-class discussion segments into resonant, experiential learning sessions for all participants — whatever their starting-point orientation.

Over and over, speed demons are exposed to — and required to notice — the kinds of majestic mathematical discoveries that are possible when you relinquish your foundational belief that only faster can be better.

And katamari experience that “slow and steady” is not an inferior way to approach mathematics but rather, a powerful orientation and set of talents that can reveal mathematical depth and structure that are hidden to the naked speed demon eye.

It also strikes me that much of this is completely at odds with the narrow-minded, and often obtuse insistence in Complex Instruction that everybody always stay together on everything, working on the “same problem” at the “same time.”

I find this obtuse because I have seen how the richness of our human experience comes from coming together and bringing our whole selves to our collaboration — not by holding ourselves back and playing small to avoid making anybody else feel less empowered.

This is what I am trying to get my learners to understand about the value of collaborating with others. Speed demons are rewarded by the mountains they climb and the spectacular landscapes this allows them to experience. Katamari are rewarded by the dazzling richness and microscopic hidden structures they discover. When we bring these experiences together and allow ourselves to share our most powerful insights, that is when we discover the full spectrum of what it means to be mathematical and to be human.

Saturday, January 30, 2016

Algebra 1 Systems of Inequalities - Dan Wekselgreene's Ohio Jones & the Templo de los Dulces treasure map

Some of Dan Wekselgreene's early puzzles, lessons, and projects are truly love poems for Algebra 1 students. And I have loved his Ohio Jones and the Templo de los Dulces systems of inequalities puzzle since the first time I read about it, did it, and used it.

So there was never any question that I would use it with my Algebra 1 students this year. The only question was, how would I make it accessible to my blind student?

Susan Osterhaus of the Texas School for the Blind and Visually Impaired has been generous beyond words with her ideas for teaching math to blind students. Her web site is filled with ideas, best practices, and links to resources for making mathematics accessible to blind students. I cannot recommend it strongly enough.

Here is how I adapted this activity:

I typed out each of the three clues as a quote by a statue — i.e., Statue 1 says...

Then I plugged each quote into an online Braille Translator (I like and downloaded the Braille text file as an image. I copied and pasted each image file onto an Omni Graffle document (though you could also use Word or Pages) next to the regular text quote. That way the student and the paraprofessional aide could easily collaborate and share information.

This took three pages, but it worked.

I traced the basic map at the bottom of the page but without all the grid lines. This became the "map" for this part of the puzzle. 

Then I copied these pages onto capsule paper and ran them through the PIAF (Pictures In A Flash) machine to create a tactile worksheet with Braille and a raised map. The PIAF machine (affectionately known around the math office as "the toaster") takes the capsule paper with all its delicious black carbon-heavy areas and raises them to create a tactile graphic that can be read by a Braille-literate blind person.

After solving the system and figuring out the target region, my student used Wikki Stix on the map to make a graph.


I traced the "big picture" outline of the map to remove as much visual noise and clutter as possible from the main image.  I added Braille labels to indicate the start and the hint at the end of the map:

I scanned this file, printed it on capsule paper, and ran it through the PIAF machine. Again, the student worked on Braille graph paper, then transferred her results to her tactile treasure map using Wikki Stix.

For each "sector" of the map, my student used Braille graph paper and Wikki Stix while her classmates used pencil and the grid on the worksheet.

It was such a joy to see her as just another team member at her table, doing mathematics and solving a puzzle. It was even more exciting to see how her table mates appreciated her mathematical skills.

All in all, a successful experience in creating an inclusive classroom!

My reduced version of the Teacher Packet (including the worksheet and instructions) plus the Braille-ready package are all on the Math Teacher's Wiki.

Wikki Stix are available in a big box on Amazon or any kids' art supply store.

Monday, January 18, 2016

Algebra 1 Inequalities – A minor 'How People Learn' unit

Here's a perfectly imperfect model unit of how I use the How People Learn stages and cycles in a typical unit in my Algebra 1 class. I'm documenting this for myself, so anybody else who finds this useful is just icing on the cake! :)

All of the files I use are in this downloadable zip file on the Math Teacher Wiki:

     Algebra 1 Inequalities unit

Here's a rough overview of how this works.

How are algebraic inequalities related to our basic number sense concepts of "more than" and "less than," and how can we use this understanding to build a more generalizeable algebraic understanding?

STAGE 1 - hands-on introductory task

(1) Deleted scene (readers' theater activity):  groups read the deleted scene about how to do Talking Points, which also contains a review of number sense concepts of more than and less than.

(2) Talking Points - set #1 — more than and less than: what does your group think?
Students follow the protocol they have just learned and do set #1 of Talking Points that active prior knowledge about which is more and which is less, given two sets of quantities.

STAGE 2 - initial provision of an expert model

Each day is different, but usually I spend about 10-15 minutes working with the whole class to "do some notes" (combination of mini-lecture and note-taking and modeling). This is often the last thing we do in the class period.

STAGE 3 - deliberate practice with metacognitive self-monitoring

Practice happens at both a macro- and a micro- level. On the micro- level, each day's homework (which is completely distinct from the day's classwork) gives a chance to practice and review. Then the first activity of the next day's class is comparing answers in your table groups and answering all questions that groups or students are able to answer for each other or for themselves. I take only Burning Questions (like only taking group questions during Complex Instruction problem-solving tasks).

At the macro-level, we are building toward two big days of Speed Dating, which is differentiated deliberate practice with metacognitive self-monitoring and peer tutoring or reteaching as needed.

We cycle back through all of these for several days, as you can see, with a new set of Talking Points each day that students have to work through and puzzle over. Each day's Talking Points build in a new piece of knowledge that is in students' Zone of Proximal Development so that they can encounter it, wrestle with it, and formalize their understanding of it. Then they get the nightly chance to practice some more.

This is a highly Vygotskian model of learning.

Eventually we need to introduce a new concept into our exploration of greater than and less than. I call this concept the concept of "betweenness." We do this through (5) another deleted scene.

Once again, we are investigating numbers and quantities, but we are extending our investigation to more abstract conceptions of quantities. We go back to Talking Points. We do some problem-solving. We struggle together. We organize our learning.

STAGE 4 - transfer task

I don't have a particularly great transfer task yet for this unit. That's why it's such a good one to use as an introduction or reintroduction to my Talking Points norms and practices.

If you have a super-terrific transfer task for Algebra 1-level linear inequalities (not yet at systems), I'm all ears. Please let us know about it in the comments section.

Let me know what you think if you try any of this!
A Postscript —
Today was the kind of day that makes all this work worth it. To review for tomorrow's unit test, we did group whiteboarding of some pretty hard problems. And even though many kids were still stumped by some of the harder problems, they felt excited. They were understanding it.

And doing it.

Doing math.

Now that is the kind of thing that makes it worth waking up at 5 in the morning five days a week for ten months of the year straight at near-poverty wages. :)

Saturday, January 16, 2016

Betweenness and non-betweenness: absolute value inequalities and Patrick Callahan

I felt a little nervous about having Patrick Callahan come to observe my classroom yesterday, but in the end, it was fun. I had asked one of our security guards, to bring him down to my room when he arrived at our school. He walked in as he always does, all mathematical open-mindedness and pedagogical curiosity.

And we got started.

I felt anxious about having him observe my conceptual lessons about betweenness and non-betweenness. I have never seen anything even close to how I understand and talk about absolute value and inequalities. I talk about boundary points and betweenness and I have students hold up their fists and point their thumbs to show me their understanding. “Is this a situation of betweenness — or NON-betweenness?” I demonstrate with my own fists, swinging my thumbs inward or outward. “Your fists are the boundary points and your thumbs are how you shade your graph on the number line. So is this a situation of betweenness... or NON-betweenness?”

If it is a situation of "betweenness," then students point their thumbs inward towards each other, touching the tips together. If it is NON-betweenness, then they point their thumbs outward in either direction, like a group of indecisive hitchhikers. And once we have done this analysis, then we can do whatever calculations we may need to find our boundary points.

So much of advanced algebra and precalculus depends on having this kind of deep conceptual understanding and thinking. Am I looking for quantities that are GREATER than...? or LESS than? Is this quantity going to be positive? or negative?

For me, the whole thing is intimately hooked together with the real number line. And with number sense. 

When we started last week, we began with an inquiry into “more than” and “less than” and widened our thinking outward from there.We connected more than and less than to number line thinking. I always emphasize Number-Line-Order and Number-Line-Thinking in my Algebra 1 classes. If they think about the number line, then they can anchor their thoughts in their bodies. LHS (or Left-Hand Side) and RHS (Right-Hand-Side) are fundamental ways of thinking in algebra. These ideas are eternal and unchanging. The number line is the foundation of everything. It gives you the “true north” of the real number system.

So we always ground our thinking in our bodies. I ask, “Left Hand Side or Right Hand Side?” “Is this a situation of betweenness or non-betweenness?” “OK, now that we know that, now what?”

I also anchor this unit in what they know about logical reasoning. They have an intuitive sense of how many possible cases a situation may present. I've been a huge Yogi Berra philosophy fan all my life, so I believe that when you come to a fork in the road, you should take it. When you come to a fork in the road, you can go left or you can go right. Or you can stay right where you are. Three possible cases. Over and over I ask them, “What’s going on here? How do you know?”

Absolute value inequalities are either situations of betweenness or situations of non-betweenness. Figure that out and then everything else will run smoothly. Then all you have to do is to use what you already know.

Once students have gotten that figured out, it’s just one more small step to combining their new knowledge with their existing knowledge. Follow the order of operations and common sense. Plus everything you know about the real number line and multiple representations. Then things can naturally unfold the right way.

But I always come back to number sense to what we know about the real number line. Numbers are the ground, the foundation.

So what Patrick walked into yesterday — this world-class mathematician and math education expert — was my bootcamp in algebraic thinking. “Hold up your fists! Is this a situtation of betweenness or non-betweenness?”  "How do you know?" And then my waiting until everybody’s thumbs are pointing in the same direction.

It is Logic 101 and numbers and anchoring our thoughts about numbers in our bodies. Like the ancient Greeks and Babylonians and Egyptians before us.

Our next step is to solidify our thinking through what How People Learn calls “deliberate practice with metacognitive awareness.” We are going to do two days of Speed Dating. Now I have to make up Speed Dating cards and a test to use on Thursday. 

And then to document my thinking.

When the class ended, Patrick came up to my tech podium and was excited. He grabbled a whiteboard marker and started sketching and pouring out ideas.

For me, that was the best possible review I could have gotten on this lesson. A five-unicorn review. A direct hit. :)

Friday, January 15, 2016

NEW STRATEGY: having students introduce themselves to Talking Points by way of a deleted scene

I wrote another one of my "deleted scenes" from various Hollywood movies as a way to get students to introduce themselves to Talking Points.

I think this is my best idea yet.

Here is a link to the deleted scene, on the Math Teacher Wiki:

                 Intro to Talking Points and inequalities - Harry Potter.pdf

Students do so much better a job of monitoring their own use of the structure. I think this makes them "own it" more.

After only two weeks, they now groan dramatically at my corny situations. But they secretly (and not-so-secretly) love it. They dive right in, choosing roles and reading dramatically.

The whole thing takes about 7 minutes.

Let me know what you think!

Wednesday, January 13, 2016

The concept of betweenness

I am coming to believe that, much like the concept of substitution, developing a deep understanding of the idea of betweenness is a huge part of the psychological and conceptual work of Algebra 1.

I have been dissatisfied for years now with the fact that we tell students about the geometric interpretation of absolute value, but we don't really get them to live it. And yet that idea of the "distance from zero" on the real number line is not something we give students time to really marinate in.

And you know what? That feels dumb to me.

So this week and part of the next, my students and I are really wallowing in that idea.

I've taken a page from my studies as a young piano student. In the study of the piano, there are certain studies of technique that really force you to slow down and take apart the finger movements. There are specific figures that you have to practice over and over and over so that they become part of your finger memory. How they feel in your fingers is how you come to relate to them.

This is not just about developing automaticity, although that is a side benefit. This is about learning to feel these foundational figures in your bones. In your body. They become so fundamental that as you learn and grow as a musician, you come to feel them when you see them coming up in a new score you are studying.

The technique does not replace musicianship. The technique supports the musicianship.

I've been noticing lately how my own experience of absolute value is about noticing boundary points at the periphery of my mathematical perception. I see them out of the corner of my mathematical mind's eye. And how an inequality is said to relate to them defines how I relate to those boundary points.

So I am taking the risk of sharing this mathematical experience with my students.

As with young piano students, we take this slowly. One figure at a time. Right now we are only dealing with the case of an absolute value being less than a nonnegative quantity. We are dealing with situations of betweenness, where an inequality presents us with a figural situation that is going to wind up with a quantity being between two boundary points.

That is all. And that is enough.

I see the effort in their faces and in their fingers as they rewrite, revise, calculate, solve, and sketch graphs. I see them noticing and wondering whether they need to use a closed dot or and open dot.

And I hear them developing the confidence that comes from experience in developing a relationship with these quantities.

They are not following rules. They are listening to their own deeper wisdom. Everybody knows something about the situation of being "between" other things. Betweenness is one of the most elemental human ideas.

They are making friends with mathematics.

Sunday, January 3, 2016

Seating charts for equity

I can't imagine traveling to new lands and not wanting to try their cuisine. But there really are people who bring their own food with them. One of the best things about traveling in my opinion is being educated  in the sense of the Latin root word — being led out of my own ignorance.

The same is true for me about attending a large, great school. It always has been. From the moment I arrive in a great new school, I feel excited and open to meeting and learning with all different kinds of people from different cultures and backgrounds. I want to expand my own limited world view.

But it seems inevitable that, without outside intervention, I often end up knowing and hanging out with the other Buddhists and Jews in any room. Cultural affinity is a force that possesses a tractor beam all its own. Fortunately, I am not the first to have noticed noticed this.

Our amazing counseling department and our Peer Resources program noticed this phenomenon too, and when they did their most recent student survey of our very large, urban, diverse student body, they put in some questions about this in their student well-being section. And the results were very moving to me.

Students overwhelmingly reported that when they first arrived at our school, they felt enormous pressure to connect with their cultural affinity groups. And for this reason, they reported, they deeply appreciate seating charts in classes that take this pressure away. This practice overwhelmingly helped them to feel that they fit in here and that those who are different from them in some ways are more like them in other ways than they are inclined to believe. It also created a zone of psychological and emotional safety to explore social connections with others not as "Others" but as fellow explorers in a safe space.

These findings touched my heart. Our kids' deeper wisdom never fail to blow me away.

So I sit here on the Sunday before the first day of Spring term making up seating charts, making sure that everybody arrives in my classes in the same boat as everybody else, and with the same opportunity to experience connection with others in as safe a space as I can create.

I will also pre-make Seating Charts #2, #3, and #4 so that it's convenient for me to change the seating without having to think. Sometimes "don't think" is the best rule.

I don't have any scintillating conclusions to draw here. I just wanted to document for myself what I am doing and why so that when I forget, I can more easily remember.